The number of midsegments in a triangle are three, which together form a
triangle inscribed in the large triangle.
Description of the relationship;
- The perimeter of a triangle is twice the perimeter of the triangle formed by the midsegments of the triangle.
Method by which the above response is derived;
Required:
The description of the relationship between the perimeter of a triangle and the perimeter of the triangle formed by its midsegments.
Solution:
According to the midsegment theorem, we have;
The length of a midsegment is half the length of the third side of a triangle.
Let ΔABC represent the given triangle and let ΔXYZ represent the length of the midsegment of the triangle, where;
X is on side [tex]\overline{AB}[/tex]
Y is on side [tex]\overline{BC}[/tex]
Z is on side [tex]\overline{AC}[/tex]
Therefore;
2 × [tex]\overline{XY}[/tex] = [tex]\mathbf{\overline{AC}}[/tex]
2 × [tex]\overline{YZ}[/tex] = [tex]\mathbf{\overline{AB}}[/tex]
2 × [tex]\overline{XZ}[/tex] = [tex]\mathbf{\overline{BC}}[/tex]
Perimeter of triangle ΔABC = [tex]\overline{AC}[/tex] + [tex]\overline{AB}[/tex] + [tex]\overline{BC}[/tex] = 2 × [tex]\overline{XY}[/tex] + 2 × [tex]\overline{YZ}[/tex] + 2 × [tex]\overline{XZ}[/tex]
Therefore;
- Perimeter of triangle ΔABC = [tex]\mathbf{2 \times \left(\overline{XY} + \overline {YZ} + \overline{XZ} \right)}[/tex]
- Perimeter of triangle ΔXYZ = [tex]\overline{XY}[/tex] + [tex]\overline{YZ}[/tex] + [tex]\overline{XZ}[/tex]
- Perimeter of triangle ΔABC = 2 × The perimeter of triangle ΔXYZ
Therefore;
- The perimeter of a triangle = Twice the perimeter of the triangle formed by the midsegments of the triangle
Learn more about the midsegment theorem here:
https://brainly.com/question/7423948
